Zeros of quadratic forms and the delta method

Abstract

This thesis presents solutions to three problems. First, we show that the optimal covering exponent for the $3$-sphere is $frac{4}{3},$ and this is joint work with T. D. Browning and R. S. Steiner. Next, we prove a result involving $h(-n)$, the class number of an imaginary quadratic field with fundamental discriminant $-n$. We give an asymptotic formula for correlations involving $h(-n)$ and $h(-n-l)$ over fundamental discriminants that avoid the congruence class $1 pmod{8}$. The result is uniform in the shift $l$, and along the way we also derive an asymptotic formula for correlations between $r_Q(n)$, the number of representations of an integer by a positive definite quadratic form $Q$. Finally, we study sums of normalised Hecke eigenvalues $lambda(n)$ of holomorphic cusp forms over thin sequences. Let $F(bs{x})$ be a diagonal quadratic form in $4$ variables, we give an upper bound for the problem of counting integer solutions of bounded height to $F(bs{x})=0$ weighted by $lambda(x_1)$, and as a consequence we derive upper bounds for certain generalised cubic divisor sums. All three problems are solved by counting integer zeros of quadratic forms using the $delta$-method.

Date of Award	25 Sept 2018
Original language	English
Awarding Institution	University of Bristol
Supervisor	Tim Browning (Supervisor)